On the Socle of Finite Primitive Permutation Groups Having Frobenious Structure

Danbaba Adamu; Momoh Umoru Sunday

doi:doi:10.11648/j.pamj.20241306.11

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On the Socle of Finite Primitive Permutation Groups Having Frobenious Structure

Danbaba Adamu^*, Momoh Umoru Sunday

Published in Pure and Applied Mathematics Journal (Volume 13, Issue 6)

Received: 7 August 2024 Accepted: 5 September 2024 Published: 13 December 2024

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Abstract

The nilpotentcy class for the Frobenius was determined based on the structure theorem. The socle of the groups were observed to be regular normal and elementary abelian such features were the conditions for the nilpotency classes, as they were the basis on which the socle of these groups constructed were nilpotent of some classes or order. The socle of the nilpotent groups whose structures is in conformity with D were classified based on the classification scheme for the finite primitive groups in relation to socle type. The socle type described in the classification scheme was in condition (1) was in line with the structure of D, as such it pave way in determining the socle with nilpotency class having same or similar structure with D. Further investigations showed that Frobenious group's were 2-transitive and the structure of D gave the conditions it being regular elementary abelian and so is nilpotent. It was observed the stabiliser of the groups in a finite primitive groups were paramount in the determination of the socle of the groups, as such much attention was given to the stabilizer of each group under consideration in a quest to determine the socle and the nilpotency class. The other conditions for the classification of finite primitive groups based of the socle type were not given much attention as it could give the needed condition for the existence of nilpotency class of the groups, as groups of such types were either almost simple, diagonal, product or twisted wreath product type. Therefore finite primitive group's under those conditions which could not give the expected nilpotency class and order were not give much attention. The degree of homogeneity was not given much priority as the article intended to discuss only the socle type and it nilpotency class or order.

Published in	Pure and Applied Mathematics Journal (Volume 13, Issue 6)
DOI	10.11648/j.pamj.20241306.11
Page(s)	79-83
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Frobenius, Groups, Socle, Nilpotency, Finite, Abelian, Regular

References

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[2]	Burnside, W (1911). Theory of groups of finite order. Cambridge University press. London.
[3]	Cameron, P. J. (1981). Finite permutation groups and finite simple groups; London Math. Soc: https://doi.org/101112/BLMS/13.1.1
[4]	Cameron, P. J. (1999). Permutation groups. Math. Soc, Student Text 45, Cambridge University Press.
[5]	Cameron, P. J. (2000). Aspect of infinite permutation groups: School of Mathematical Sciences, Queen Mary, University of London.
[6]	Cameron, P. J. (2000). t-orbit homogenous permutations. London Math Soc. Subject Classification 20B10, https://doi.org/10.1112/Sooooooooooooooo
[7]	Cameron, P. J. (2004). The encyclopaedia of design theory on primitive permutation groups. www.maths.qmul.ac.uk Retrieved on 12/07/2015.
[8]	Cameron, P. J (2012). Permutation groups and regular semi groups. London Math Subject classification. 1273-1285.
[9]	Cameron, P. J. (2013) Permutation group and transformation semi-group, Novi sad Algebraic Conference. University of St. Andrews.
[10]	Dixion, J., & Mortimer, B (1996). Permutation groups Graduate texts in mathematics. Springer New York`.
[11]	Frobenius, F. G. (1904). Uber die charaktere der mehch trasitiven gruppen. mouton de grupter 558-571. Berlin.
[12]	Kantor, W. M. (1972). k-Homogeneous permutation groups. Math Z, 124, 261-265. Springer Verlag.
[13]	Khukhr, O (2013). Finite p-groups with a Frobenius group of automorphism whose kernel is cyclic p-group. Manchester institute for Mathematical Sciences: School of Mathematics, University of Manchester.
[14]	Liebeck, W., & Praeger C. E, & Sax, I. J. (2000). Primitive permutation groups with a regular subgroup. Journal of Mathematics subject classification. 20B15, 05C25.
[15]	Livingstone, D., & Wagner, A. (1965). Transitivity of finite permutation groups on unordered sets. Math Z. 90 393-403.
[16]	Martin, W. J., & Segan, B (2000). New notion of transitivity for groups and set of permutations. Journal of Math Soc. Math Subject Classification 20B20.
[17]	O′Nan, M. E., & Scott, L (1979). Finite groups. Santa Cruz conference. London Math Soc (2): 32.
[18]	Wielandt, H. (1969). Permutation groups through the invariant relation and invariant functions. Columbus. 399514.

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APA Style

Adamu, D., Sunday, M. U. (2024). On the Socle of Finite Primitive Permutation Groups Having Frobenious Structure. Pure and Applied Mathematics Journal, 13(6), 79-83. https://doi.org/10.11648/j.pamj.20241306.11

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Adamu, D.; Sunday, M. U. On the Socle of Finite Primitive Permutation Groups Having Frobenious Structure. Pure Appl. Math. J. 2024, 13(6), 79-83. doi: 10.11648/j.pamj.20241306.11

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AMA Style

Adamu D, Sunday MU. On the Socle of Finite Primitive Permutation Groups Having Frobenious Structure. Pure Appl Math J. 2024;13(6):79-83. doi: 10.11648/j.pamj.20241306.11

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@article{10.11648/j.pamj.20241306.11,
  author = {Danbaba Adamu and Momoh Umoru Sunday},
  title = {On the Socle of Finite Primitive Permutation Groups Having Frobenious Structure
},
  journal = {Pure and Applied Mathematics Journal},
  volume = {13},
  number = {6},
  pages = {79-83},
  doi = {10.11648/j.pamj.20241306.11},
  url = {https://doi.org/10.11648/j.pamj.20241306.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20241306.11},
  abstract = {The nilpotentcy class for the Frobenius was determined based on the structure theorem. The socle of the groups were observed to be regular normal and elementary abelian such features were the conditions for the nilpotency classes, as they were the basis on which the socle of these groups constructed were nilpotent of some classes or order. The socle of the nilpotent groups whose structures is in conformity with D were classified based on the classification scheme for the finite primitive groups in relation to socle type. The socle type described in the classification scheme was in condition (1) was in line with the structure of D, as such it pave way in determining the socle with nilpotency class having same or similar structure with D. Further investigations showed that Frobenious group's were 2-transitive and the structure of D gave the conditions it being regular elementary abelian and so is nilpotent. It was observed the stabiliser of the groups in a finite primitive groups were paramount in the determination of the socle of the groups, as such much attention was given to the stabilizer of each group under consideration in a quest to determine the socle and the nilpotency class. The other conditions for the classification of finite primitive groups based of the socle type were not given much attention as it could give the needed condition for the existence of nilpotency class of the groups, as groups of such types were either almost simple, diagonal, product or twisted wreath product type. Therefore finite primitive group's under those conditions which could not give the expected nilpotency class and order were not give much attention. The degree of homogeneity was not given much priority as the article intended to discuss only the socle type and it nilpotency class or order.
},
 year = {2024}
}

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TY  - JOUR
T1  - On the Socle of Finite Primitive Permutation Groups Having Frobenious Structure

AU  - Danbaba Adamu
AU  - Momoh Umoru Sunday
Y1  - 2024/12/13
PY  - 2024
N1  - https://doi.org/10.11648/j.pamj.20241306.11
DO  - 10.11648/j.pamj.20241306.11
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 79
EP  - 83
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20241306.11
AB  - The nilpotentcy class for the Frobenius was determined based on the structure theorem. The socle of the groups were observed to be regular normal and elementary abelian such features were the conditions for the nilpotency classes, as they were the basis on which the socle of these groups constructed were nilpotent of some classes or order. The socle of the nilpotent groups whose structures is in conformity with D were classified based on the classification scheme for the finite primitive groups in relation to socle type. The socle type described in the classification scheme was in condition (1) was in line with the structure of D, as such it pave way in determining the socle with nilpotency class having same or similar structure with D. Further investigations showed that Frobenious group's were 2-transitive and the structure of D gave the conditions it being regular elementary abelian and so is nilpotent. It was observed the stabiliser of the groups in a finite primitive groups were paramount in the determination of the socle of the groups, as such much attention was given to the stabilizer of each group under consideration in a quest to determine the socle and the nilpotency class. The other conditions for the classification of finite primitive groups based of the socle type were not given much attention as it could give the needed condition for the existence of nilpotency class of the groups, as groups of such types were either almost simple, diagonal, product or twisted wreath product type. Therefore finite primitive group's under those conditions which could not give the expected nilpotency class and order were not give much attention. The degree of homogeneity was not given much priority as the article intended to discuss only the socle type and it nilpotency class or order.

VL  - 13
IS  - 6
ER  -

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Author Information

Danbaba Adamu

Department of Mathematics, University of Jos, Jos, Nigeria

Contact Email
Momoh Umoru Sunday

Department of Mathematics, University of Jos, Jos, Nigeria

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Plain Text BibTeX RIS

APA Style

Adamu, D., Sunday, M. U. (2024). On the Socle of Finite Primitive Permutation Groups Having Frobenious Structure. Pure and Applied Mathematics Journal, 13(6), 79-83. https://doi.org/10.11648/j.pamj.20241306.11

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ACS Style

Adamu, D.; Sunday, M. U. On the Socle of Finite Primitive Permutation Groups Having Frobenious Structure. Pure Appl. Math. J. 2024, 13(6), 79-83. doi: 10.11648/j.pamj.20241306.11

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AMA Style

Adamu D, Sunday MU. On the Socle of Finite Primitive Permutation Groups Having Frobenious Structure. Pure Appl Math J. 2024;13(6):79-83. doi: 10.11648/j.pamj.20241306.11

Copy | Download

@article{10.11648/j.pamj.20241306.11,
  author = {Danbaba Adamu and Momoh Umoru Sunday},
  title = {On the Socle of Finite Primitive Permutation Groups Having Frobenious Structure
},
  journal = {Pure and Applied Mathematics Journal},
  volume = {13},
  number = {6},
  pages = {79-83},
  doi = {10.11648/j.pamj.20241306.11},
  url = {https://doi.org/10.11648/j.pamj.20241306.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20241306.11},
  abstract = {The nilpotentcy class for the Frobenius was determined based on the structure theorem. The socle of the groups were observed to be regular normal and elementary abelian such features were the conditions for the nilpotency classes, as they were the basis on which the socle of these groups constructed were nilpotent of some classes or order. The socle of the nilpotent groups whose structures is in conformity with D were classified based on the classification scheme for the finite primitive groups in relation to socle type. The socle type described in the classification scheme was in condition (1) was in line with the structure of D, as such it pave way in determining the socle with nilpotency class having same or similar structure with D. Further investigations showed that Frobenious group's were 2-transitive and the structure of D gave the conditions it being regular elementary abelian and so is nilpotent. It was observed the stabiliser of the groups in a finite primitive groups were paramount in the determination of the socle of the groups, as such much attention was given to the stabilizer of each group under consideration in a quest to determine the socle and the nilpotency class. The other conditions for the classification of finite primitive groups based of the socle type were not given much attention as it could give the needed condition for the existence of nilpotency class of the groups, as groups of such types were either almost simple, diagonal, product or twisted wreath product type. Therefore finite primitive group's under those conditions which could not give the expected nilpotency class and order were not give much attention. The degree of homogeneity was not given much priority as the article intended to discuss only the socle type and it nilpotency class or order.
},
 year = {2024}
}

Copy | Download

TY  - JOUR
T1  - On the Socle of Finite Primitive Permutation Groups Having Frobenious Structure

AU  - Danbaba Adamu
AU  - Momoh Umoru Sunday
Y1  - 2024/12/13
PY  - 2024
N1  - https://doi.org/10.11648/j.pamj.20241306.11
DO  - 10.11648/j.pamj.20241306.11
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 79
EP  - 83
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20241306.11
AB  - The nilpotentcy class for the Frobenius was determined based on the structure theorem. The socle of the groups were observed to be regular normal and elementary abelian such features were the conditions for the nilpotency classes, as they were the basis on which the socle of these groups constructed were nilpotent of some classes or order. The socle of the nilpotent groups whose structures is in conformity with D were classified based on the classification scheme for the finite primitive groups in relation to socle type. The socle type described in the classification scheme was in condition (1) was in line with the structure of D, as such it pave way in determining the socle with nilpotency class having same or similar structure with D. Further investigations showed that Frobenious group's were 2-transitive and the structure of D gave the conditions it being regular elementary abelian and so is nilpotent. It was observed the stabiliser of the groups in a finite primitive groups were paramount in the determination of the socle of the groups, as such much attention was given to the stabilizer of each group under consideration in a quest to determine the socle and the nilpotency class. The other conditions for the classification of finite primitive groups based of the socle type were not given much attention as it could give the needed condition for the existence of nilpotency class of the groups, as groups of such types were either almost simple, diagonal, product or twisted wreath product type. Therefore finite primitive group's under those conditions which could not give the expected nilpotency class and order were not give much attention. The degree of homogeneity was not given much priority as the article intended to discuss only the socle type and it nilpotency class or order.

VL  - 13
IS  - 6
ER  -

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